Motivic cohomology groups are isomorphic to higher Chow groups in any characteristic.
type: Item
instance of: scholarly article
zbMATH Open document ID: 1057.14026
DOI: 10.1155/S107379280210403X
author: Vladimir Voevodsky
publication date: 2002-06-23T00:00:00Z
Mathematics Subject Classification ID: 14F42
Mathematics Subject Classification ID: 14C15
Mathematics Subject Classification ID: 19E15
zbMATH DE Number: 1724017
zbMATH Keywords: algebraic \(K\)-theory
zbMATH Keywords: derived categories
zbMATH Keywords: sheaves with transfer
MaRDI profile type: Publication
full work available at URL: s107379280210403x
OpenAlex ID: W2307632517
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published in: IMRN. International Mathematics Research Notices
title: Motivic cohomology groups are isomorphic to higher Chow groups in any characteristic.
review text: In this short research notice, the author shows that two differently defined types of motivic cohomology groups for smooth schemes over any field are in fact isomorphic. On the one hand, \textit{E. M. Friedlander} and \textit{A. Suslin} recently gave a construction of a certain motivic cohomology theory that was directly related, through a spectral sequence, to the higher \(K\)-theory of Chow groups [Ann. Sci. Éc. Norm. Supér., IV. Sér. 35, No. 6, 773--875 (2002; Zbl 1047.14011)]. Another kind of motivic cohomology was introduced by \textit{A. Suslin} and the author of the present note in order to approach the so-called Bloch-Kato conjecture [in: The arithmetic and geometry of algebraic cycles. Proceedings of the NATO Advanced Study Institute, Banff, Canada, Jun 7--19, 1998. Math. Phys. Sci. 548, 117--189 (2000; Zbl 1005.19001)]. The proof that these two kinds of motivic cohomology are equivalent, which is the main result of the present note, implies that the Suslin-Voevodsky motivic cohomology groups of a smooth scheme over any field are isomorphic to certain higher Chow groups. More precisely, there is a natural isomorphism NEWLINE\[NEWLINEH^{p,q}(X,\mathbb{Z})\cong CH^q(X, 2q- p),NEWLINE\]NEWLINE and the same identity holds for arbitrary coefficients on both sides.
reviewed by: Werner Kleinert
description: scientific article; zbMATH DE number 1724017